The SU(1,1)/U(1) dynamical symmetry of a family of particles in the field of a Kerr black hole

TL;DR

The paper demonstrates that particles moving around a Kerr black hole exhibit SU(1,1)/U(1) symmetry, enabling a unified time variable and harmonic oscillator form for their Hamiltonian, which aids in quantization.

Contribution

It reveals a new dynamical symmetry in particle motion near Kerr black holes and introduces a global time variable for quantization.

Findings

Identification of SU(1,1)/U(1) symmetry in particle dynamics

Establishment of a global time variable shared by all particles

Reformulation of Hamilton's equations as harmonic oscillators

Abstract

A family of particles moving within a cone centered on a Kerr black hole is shown to have SU(1,1)/U(1) dynamical symmetry. This symmetry is used to identify a global time variable shared by all particles in the family. With this time variable, Hamilton's equations for the family of particles have the canonical form of the harmonic oscillator. The SU(1,1)/U(1) dynamical symmetry, along with the well defined global time variable and observer, assists in determining the quantization of the motion.